We introduce the matrix-valued time-varying Main Effects Factor Model (MEFM). MEFM is a generalization to the traditional matrix-valued factor model (FM). We give rigorous definitions of MEFM and its identifications, and propose estimators for the time-varying grand mean, row and column main effects, and the row and column factor loading matrices for the common component. Rates of convergence for different estimators are spelt out, with asymptotic normality shown. The core rank estimator for the common component is also proposed, with consistency of the estimators presented. We propose a test for testing if FM is sufficient against the alternative that MEFM is necessary, and demonstrate the power of such a test in various simulation settings. We also demonstrate numerically the accuracy of our estimators in extended simulation experiments. A set of NYC Taxi traffic data is analysed and our test suggests that MEFM is indeed necessary for analysing the data against a traditional FM.

翻译：本文提出了矩阵值时变主效应因子模型（MEFM）。MEFM是对传统矩阵值因子模型（FM）的一种推广。我们给出了MEFM及其可识别性的严格定义，并提出了时变总均值、行主效应、列主效应以及公共成分的行与列因子载荷矩阵的估计量。我们阐明了不同估计量的收敛速率，并证明了其渐近正态性。同时，提出了公共成分的核心秩估计量，并给出了估计量的一致性证明。我们设计了一种检验，用于检验FM是否充分，其备择假设为MEFM是必要的，并在多种模拟设置中展示了该检验的功效。通过扩展的模拟实验，我们也数值地证明了所提估计量的准确性。最后，我们分析了一组纽约市出租车流量数据，我们的检验表明，相对于传统FM，MEFM对于分析该数据确实是必要的。